Energy absorption during compression and impact of dry

Granular Matter (2010) 12:15–47DOI 10.1007/s10035-009-0161-3

Energy absorption during compression and impact of dryelastic-plastic spherical granules

Sergiy Antonyuk · Stefan Heinrich · Jürgen Tomas ·Niels G. Deen · Maureen S. van Buijtenen ·J. A. M. Kuipers

Received: 15 July 2008 / Published online: 13 January 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract The discrete modelling and understanding of theparticle dynamics in fluidized bed apparatuses, mixers, millsand others are based on the knowledge about the physicalproperties of particles and their mechanical behaviour dur-ing slow, fast and repeated stressing. In this paper modelparameters (modulus of elasticity, stiffness, yield pressure,restitution coefficient and strength) of spherical granules(γ-Al2O3, zeolites 4A and 13X, sodium benzoate) with dif-ferent mechanical behaviour have been measured by singleparticle compression and impact tests. Starting with the elas-tic compression behaviour of granules as described by Hertztheory, a new contact model was developed to describe theforce-displacement behaviour of elastic-plastic granules. Theaim of this work is to understand the energy absorption dur-ing compression (slow stressing velocity of 0.02 mm/s) andimpact (the impact velocity of 0.5–4.5 m/s) of granules. Forall examined granules the estimated energy absorption dur-ing the impact is found to be far lower than that during com-pression. Moreover, the measured restitution coefficient isindependent of the impact velocity in the examined rangeand independent of the load intensity by compression (i.e.maximum compressive load). In the case of repeated load-ing with a constant load amplitude, the granules show cyclic

S. Antonyuk (B) · S. HeinrichInstitute of Solids Process Engineering and Particle Technology,Hamburg University of Technology, Denickestr. 15,21073 Hamburg, Germanye-mail: [email protected]

J. TomasInstitute of Process Engineering, Otto-von-Guericke-Universityof Magdeburg, Universitaetsplatz 2, 39106 Magdeburg, Germany

N. G. Deen · M. S. van Buijtenen · J. A. M. KuipersInstitute for Mechanics Processes and Control Twente, Universityof Twente, PO Box 217, 7500 AE Enschede, The Netherlands

hardening with increasing restitution coefficient up to a cer-tain saturation in the plastic deformation. A model was pro-posed to describe the increase of the contact stiffness with thenumber of cycles. When the load amplitude is subsequentlyincreased, further plastic deformation takes place and therestitution coefficient strongly decreases.

Keywords Granules · Energy absorption · Elastic-plasticbehaviour · Restitution coefficient · Contact model

List of symbols

d1 Diameter of the granule (m)dk,pl Diameter of plastic deformed contact area (m)E∗ Average modulus of elasticity

of contact partners (N/m2)E Modulus of elasticity (N/m2)F Normal force (N)FB Breakage force (N)Fcyc, FU Maximum force during a loading cycle (N)kn Contact stiffness in the normal direction (N/m)

of contact partners (N/m2)e Restitution coefficienteeq Equivalent restitution coefficientk∗

n The median contact stiffnessin normal direction (N/m)

m1 Mass of the granule (kg)pF Plastic micro-yield strength of granule

contact (N/m2)R∗ Median radius of contact partners (m)R1 Radius of the granule (m)rk Radius of the contact area (m)s Displacement (m)

123

16 S. Antonyuk et al.

t Time (s)v1 Impact velocity (m/s)v1,F Minimum impact velocity which corresponds

to the yield point (m/s)vB Stressing velocity during compression test (m/s)W Loading energy (J)Wkin Kinetic energy (J)

Greek lettersκA Elastic-plastic contact area coefficientν Poisson ratio� Impact angle (deg)ρ1 Density of the granule (kg/m3)ρ f Air density (kg/m3)σmax Strength of dominantly plastic granules (N/m2)ωR Angular velocity of the granule after rebound ( rad/s)

Subscripts

B Breakageel Elasticel-pl Elastic-plasticF Yield point, flowL Loadingmax Maximumn NormalR Restitution, reboundpl Plastict Tangentialtot TotalU Unloadingz Current number of the cycle1 and g Granule2 and w Wall

1 Introduction

Many products, e.g. catalysts, adsorbents, fertilizers, tabletsetc., are often produced as granules. Obvious advantages ofthe granules in comparison to powders are higher packingdensity, better flow behaviour as well as less dust formation.Due to granulation some technological problems, like longtime consolidation or segregation of the bulk materials inbunkers and transport containers can be avoided. Moreover,desirable properties such as a regular shape, chemical com-position, narrow particle size distribution, porosity, internalsurface can be obtained.

During the processing sequence, transportation and han-dling the granules are exposed to a lot of mechanical stressingdue to granule-granule and granule-apparatus wall impacts(Fig. 1). Stressing intensity and frequency can be set equiva-lent to the conditions occurring in processing apparatus likedryers, conveyors, granulators or mixers.

The mechanical interactions of the granular assemblyduring a process can be studied by numerical simulationwith the discrete element method (DEM) or discrete particlemethod (DPM). In the last ten-fifteen years, these methodsfound many applications in the mechanics of bulk materials,i.e. flow behaviour and discharge of silos [8,49,50], fillingand packing [111], shearing [25,35,47,61,103], heap forma-tion [13,36,65], bed configuration and loading conditionsin fall mills [63,69,105], compression behaviour [68,78],mixing and transportation [12,27,44,52,54,107], fluidisedbed processes [31,40,41,58,66,90,102,104,110,112], heatexchange processes [41,50].

In these simulations, the mechanical interactions betweenparticles as well as between particle and apparatus wallsare calculated. These include the force and stress distribu-tions within the bulk material, velocities and trajectories ofindividual particles. From the simulation, important processparameters are determined, like e.g. the residence time of amaterial in the apparatus during filling and discharge of bun-kers as well as mixing or drying. Based on the special contactmodels the complicated behaviour of the cohesive and wetbulk materials can be predicted [25,103]. These simulationsprovide physical knowledge about the micro-macro interac-tions of particles and macroscopic particle bed behaviour,which can be used to achieve understanding and solution offrequently occurring breakdowns in the process due to flowdisturbances and unstable mass flows.

Through DEM, the individual granules can be modelled asa realistic mechanical system of primary particles which arebonded by adhesion forces [6]. Hence, the important influ-ences of the microstructure concerning porosity, density, pri-mary particle distribution, stiffness of the primary particles,their elasticity or plasticity as well as binder properties ofthe granule on deformation and breakage behaviour can beexamined. Many interesting results of single granule simu-lations have been reported in the literature [6,26,78,82,95].A new stage of modelling should be the simulation of anassembly of granules subjected to different stressing condi-tions. Hassanpour et al. [29] describe the deformation andbreakage of an agglomerate within a bed of particles, sub-jected to shearing. Grof et al. [24] performed the simulationsof uniaxial compression tests of needle-shaped agglomer-ates within a random packed bed. However, no simulationof a bed with granules containing many primary particles inmicrometer size range has been provided. This is becauseof the large computational effort that is needed to simu-late a very high number of small particles. The time stepof DEM calculations that ensures numerical stability is pro-portional to the mass of the smallest particle in the sys-tem [6]. Therefore in the DEM/DPM-models the granulesare usually represented as homogeneous particles [31,40,58,66,67,78,110] and the influence of their microstructure isneglected.

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Energy absorption during compression and impact of dry elastic-plastic spherical granules 17

Fig. 1 Typical stressing ofparticles during agglomerationin a rotated drum (a),granulation in fluidized bed (b),transportation (c) and discharge(d)

Fig. 2 Different loading modes of a granule: a impact on the one target,b compression between two tools with two contacts

Based on the assumption of a continuous matter, contacttheory and continuum mechanics can be used to describe themechanical behaviour of a granule without breakage throughexperimentally observed mechanical parameters like modu-lus of elasticity, yield point, contact stiffness. In our previ-ous publications [5,6] as well as in numerous experimentalstudies of other authors [10,21,64,71,78,84], it was shownthat the granules, similar to the solids, have different con-tact behaviour from dominantly elastic to dominantly plasticwhich is often time and deformation rate dependent and isrelated to adhesion effects.

In this work, we consider the behaviour of spherical gran-ules by slow compression (vB = 0.02mm/s) and rapidimpact stressing (vB = 0.5–4.5 m/s) in normal direction(Fig. 2). It is assumed that the granule is homogeneous andisotropic.

One objective is to describe the different force-displace-ment behaviour obtained by compression tests to determinethe mechanical properties concerning the contact stiffness,modulus of elasticity, yield point as well as the restitu-tion coefficient. The second objective is to compare theenergy absorption of granules during slow compressionand impact where the period of wall-granule interactionis very short. If the energy absorption in both stress-ing modes, at the same energy input, is nearly equalthe data observed by simple compression test could beapplied to modelling of impact stressing. The results willbe used for the selection of an adequate contact modeland for its following implementation in DPM for the mod-elling of particle collision and agglomeration in fluidizedbeds.

This paper is organized as follows: first, the contact mod-els for elastic, elastic-plastic and plastic spherical particlesby slow compression and impact are described. Based onthese approaches the energy balance of granule loading isdefined. In this work the dissipation energy in this balanceis estimated using the restitution coefficient. Therefore, firstthe experimental methods and devices to measure this ener-getic parameter are described. Then, the different effects,which can influence the restitution coefficient, are discussedand a classification of the models to predict these effects iscarried out. In the third chapter the materials and experimen-tal setups used in this work are described. The experimen-tal part consists on compression (slow loading) and free-falltests (rapid stressing) which give the physical understand-ing of the energy absorption of the granules with differentmaterial behaviour. Due to compression tests the importanteffect of the repeated loading on the material parameters ofgranules is obtained. Concluding the experimental resultsthe energy absorption of granules during compression andimpact is compared.

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Fig. 3 Characteristic pressure distribution p(rk ) in a wall-granulecontact during elastic deformation

2 Theoretical approaches

2.1 Characteristics of elastic contact deformationaccording to Hertzian theory

During uniaxial loading of a comparatively soft sphericalgranule with a smooth stiff wall (flat surface), the contact areadeforms as a circle (see Fig. 3). The contact radius, force andinternal pressure distribution depend on the granule radius,stiffness and deformation behaviour (elastic, elastic-plasticor plastic) of the two contacting materials.

The pioneering work of Hertz [30] describes the ellipticalpressure distribution pel(rk)in the circular contact area of aradius rk,el (Fig. 3):

(pel

pmax

)2

= 1 −(

rk

rk,el

)2

, rk ≤ rk,el . (1)

The tensile region at the perimeter of the contact pressuredistribution is responsible for bending [89]. Therefore, theradius of the totally deformed area is larger than the contactradius: rk,max ≥ rk,el (Fig. 3).

Hertz also found that the maximum contact pressure inthe centre of the contact area is given by:

pmax = 3Fel

2πr2k,el

, (2)

with a radius:

rk,el =(

3R∗Fel

2E∗

)1/3

. (3)

The effective modulus of elasticity E∗ of contact partner(E2 >> E1, E2 → ∞) is given as:

E∗ = 2

(1 − ν2

1

E1+ 1 − ν2

2

E2

)−1

≈ 2

1 − ν21

E1, (4)

where ν is Poisson ratio. Note that we use index 1 for thegranule and index 2 for the wall.

The median radius of contact partners R∗ is a character-istic radius of contact surface curvature:

R∗ =(

1

R1+ 1

R2

)−1

≈ R1. (5)

The relation between elastic contact force and displacementin normal direction is non-linear as found by Hertz [30] forthe point loading (Fig. 2a):

Fel = 2

3E∗√R∗s3 ≈ 4

3

E1

1 − v21

√d1

2s3, (6)

where d1 is the granule diameter. s is the full elastic dis-placement composed of the contact and of the whole sphere[30]:

s = r2k,el/R∗. (7)

Due to the parabolic curvature F(s), the median contact stiff-ness in normal direction increases with increase in displace-ment s and granule diameter d1 [97]:

k∗n,el = dFel

ds= E∗√R∗s ≈

√2

1 − v2 E1

√d1s. (8)

By impact stressing the median contact stiffness k∗n,el equals

the granule stiffness kn,el,g since the granule is much softerthan the wall (kn,el,w):

k∗n,el =

(1

kn,el,g+ 1

kn,el,w

)−1

≈ kn,el,g. (9)

During compression with two plates (Fig. 2b) two contactareas are formed. In this case the force-displacement rela-tionship for the compression can be given in analogy withEq. (6) as:

Fel,w−g−w = 1

6E∗√d1s3 ≈ 1

3

E1(1 − v2

1

)√d1s3. (10)

Here, the displacement s is the sum of the displacements ofboth contacts and of the whole granule (s := s/2).

The average elastic stiffness of this wall-granule-wall-sys-tems results from sequence contacts, in Eq. (11). Hence, the

mean contact stiffness(

k∗n,el,w−g−w

), usually measured by a

compression test, equals half the stiffness of granule (kn,el,g):

k∗n,el,w−g−w = dFel,w−g−w

ds

= 1

4E∗√d1s ≈ 1

2(1 − v2

1

) E1

√d1s. (11)

123


Using Eq. (6) the work of elastic deformation during theimpact can be written as:

Wel =s∫

0

Felds = 4

15E∗√R∗s5 = 2

5Fels. (12)

The law of energy conservation during the elastic impact isgiven as:

m∗

2

(ds

dt

)2

+ 4

15E∗√R∗s5 = m∗

2ν∗2, (13)

where m∗ is effective mass of both contact partners and v∗ istheir relative impact velocity.

m∗ =(

1

m1+ 1

m2

)−1

. (14)

The maximum of the force Fel,max and overlap sel,max, whichare reached at zero relative velocity, as well as the time ofimpact tel , i.e. duration of contact, can be derived from theenergy balance in Eq. (13) [56]:

sel,max =(

225

64

m∗2ν∗4

E∗2 R∗

)1/5

, sel,max ∼ V ∗ 4/5, (15)

Fel,max =(

125m∗3 E∗2 R∗ν∗6

144

)1/5

, Fel,max ∼ V ∗ 6/5,

(16)

tel = 3.78

(m∗2

ν∗ R∗E∗2

)1/5

, tel ∼ V ∗−1/5. (17)

The Hertzian contact theory of the elastic spheres has cer-tain restrictions regarding description of a dynamic collision[33]. The first condition of this theory can be formulatedas follows: the contact pressure must increase slow enoughto consider the quasi-static conditions. Secondly, the impacttime must be longer than the propagation time of releasedelastic waves along the whole length of each body impacted,Eq. (18) [48]. In this case, their effect on the stress state isnegligible.

tel

tw>> 1. (18)

If expression (18) does not hold, the stress state is a result ofadvancing elastic or elastic-plastic waves [1,108]. The prop-agation time of the elastic wave into a contact partner i alongthe normal impact can be given as:

tw = di

vw

= di

(ρi

Ei

)1/2

, (19)

where vw is wave propagation velocity.Substituting the impact time according to Eq. (17) and the

propagation time according to Eq. (19) into Eq. (18) givesthe relative critical impact time:

tel,cri t = tel

tw= 3.78

(m∗2

ν∗ R∗E∗2

)1/5(

Ei

4R2i ρi

)1/2

, (20)

where ρi is the density of the contact partner i.

2.2 Description of elastic-plastic contact deformation

Because of the high pressure within the granule contact area(100–700 MPa, see experiments described below) the yieldlimit of the granule can be reached before the breakage. Theplastic deformation starts when the maximum pressure inthe centre of the contact reaches the micro-yield strength(p = pmax = pF ). Because of a confined stress field themicro-yield strength pF is higher than the macroscopic yieldstress for tension σF,t (Molerus 1975; Pisarenko 1986). Thevalue pF was found by Davis [17] as 1.59×σF,t . We assumefor our granules that the pressure pF within the plastic contactarea is a constant and that is deformation rate independent,i.e. the hardening effect is neglected.

When the granule diameter decreases, the contact pressureincreases (see Eqs. 2 and 3). Thus at constant force the smallergranules behave softer than larger particles. The force-displacement curves measured in this work confirm the elas-tic-plastic behaviour of the granules beyond the yield point.In this case, neither the elastic nor the plastic contribution tothe total deformation can be neglected.

The schematic representation of the applied mechanicalmodel and the corresponding force-displacement curve inFig. 4 explain the elastic-plastic behaviour. From the begin-ning of loading up to the yield point (F < FF ) the force-displacement curve shows a nonlinear elastic deformationof sphere according to Hertz law. When the yield pointis reached(F ≥ FF ) plastic deformation follows, whichis additionally affected by continuous elastic deformation(Fel,y in Fig. 4a). The resulting normal contact force duringelastic-plastic deformation is represented as superimposedspring and slider elements (Fig. 4a):

F = Fel,y + Fpl = Fel . (21)

The plastic deformation is close to the contact centre of thecircular contact area (rk ≤ rk,pl in Fig. 5) and elastic defor-mation is generated at the limit of the contact (ring rk > rk,pl)

where the pressure is smaller than the yield point.Both areas increase with increasing deformation during

the loading. An elastic-plastic contact area coefficient κA

represents the ratio of plastic contact deformation area Apl

to total contact deformation area AK = Ael + Apl , see Tomas[97]:

κA = 2

3+ 1

3

Apl

Ak, (22)

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Fig. 4 Schematic of elastic-plastic contact model (a) for point loadingin normal direction and corresponding force-displacement curve b

Fig. 5 Characteristic pressure distribution p(rk) in a wall-granulecontact during elastic-plastic deformation

which after substituting the corresponding contact areas maybe written as:

κA = 1 − 1

33

√sF

s, (23)

where sF is the contact displacement at yield point. Eq. (23) isapplicable to describe both compression and impact loadingmodes.

The ratio Apl/Ak is 0 for perfectly elastic (Apl → 0) and1 for perfectly plastic deformation (Ael → 0). Thus the areacoefficient κAlies in the range between 2/3 for ideally elasticbehaviour to 1 for ideally plastic contact deformation.

The force-displacement relationship for elastic-plasticdeformation (s ≥ sF ) during the point loading of the gran-ule can be derived from the model of Tomas [97,103] byneglecting the adhesion in the contact (κp = 0 in [97]):

Fel−pl = πr2k pFκA. (24)

Substituting Eqs. (7) and (23) into Eq. (24) we obtained thefollowing expression:

Fel−pl = π R1 pF

(1 − 1

33

√sF

s

)s. (25)

The force-displacement curve at yield point (s = sF ) changesfrom the elastic curve to elastic-plastic yield limit. Therefore,the Eqs. (25) and (6) should give equal values of the forceat the yield point: Fel(sF ) = Fel−pl(sF ). Hence, the micro-yield strength and the displacement at yield point can bederived [94,96]:

pF = E∗

π

√sF

R1, (26)

sF = π2 R1 p2F

E∗2 . (27)

The force at yield point can be found, substituting Eq. (27)into Eq. (6):

FF = 2π3 R21 p3

F

3E∗2 . (28)

From Eqs. (3) and (28) the contact radius at the beginning ofplastic yielding (s = sF ) may be written as [94]:

rk,F = π R1 pF

E∗ . (29)

The stiffness during the elastic-plastic displacement is pro-portional to the radius of the granule and micro-yieldstrength:

k∗n,el−pl = dFel−pl

ds= π R1 pF

(1 − 2

93

√sF

s

). (30)

The elastic-plastic stiffness increases with increasing thedisplacement s. Due to change in the force-displacementcurve from the elastic to the plastic range the stiffnesses ofelastic and elastic-plastic displacement in the yield point arenot equal. Their ratio at the beginning of yielding may beobtained from Eqs. (10), (25), (26) and (30), as follows:

k∗n,el−pl (sF )

k∗n,el (sF )

=(1 − 2

9

)π R1 pF

E∗ (R1sF )12

=(1 − 2

9

)E∗ (R1sF )

12

E∗ (R1sF )12

= 7

9. (31)

In the case of uniaxial compression, the half of the totaldisplacement must be considered in Eqs. (25) and (26) tocalculate the micro-yield strength from the measured force-displacement curve:

Fel−pl,w−g−w = π R1 pF

(1 − 1

33

√sF

s

)s

2. (32)

2.3 Dominantly plastic contact deformation of granules

For dominantly plastic granules, the elastic range is muchsmaller in comparison with the plastic displacement. Thus,it can be assumed that the whole contact area deforms plasti-cally and no surface deformation outside of the contact zone

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Fig. 6 Contact geometry by perfectly plastic deformation of the spher-ical granule

occurs. In case of point loading (Fig. 6a), the contact radiusis given by:

r2k,pl = R2

1 − (R1 − s)2 = 2R1s − s2 ≈ d1s. (33)

Assuming that the plastic displacements at the two contactsare equal (sK ,1 = sK ,2, s = sK ,1 +sK ,2, and rk,pl,1 = rk,pl,2

in Fig. 6b), the contact radius by compression may beobtained as:

r2k,pl,w−g−w = R1s. (34)

To model the plastic behaviour only a slider element is usu-ally used [113]. The repulsive force against plastic displace-ment of the compressed granule is calculated as:

Fpl,w−g−w = pF AK = πr2k,pl pF = π R1 pF s (35)

The contact stiffness is constant for a perfectly plastic yield-ing material:

k∗n,pl,w−g−w = dFpl,w−g−w

ds= π R1 pF . (36)

2.4 Energy balance of granule loading

Based on the considerations for the deformation behaviourof a compressed granule given before, the energy balance ofthe loading can be written in the general form as:

WL = Wel,el + Wel−pl + Wdiss,add = Wel,el + Wel,el−pl

+Wpl,el−pl + Wdiss,add, (37)

where WL is the full deformation work or energy absorptionduring the compression. Wel,el is the elastic strain energydone up to yield point. Wel,el−pl and Wpl,el−pl are elasticand plastic strain energy contribution, respectively, occurringduring elastic-plastic deformation. Wdiss,add is the energyloss due to friction which is mostly converted into heat.

The individual terms of Eq. (37) can be obtained by inte-grating the corresponding force-displacement relationshipsin Eqs. (10) and (32):

Wel,el =SF∫

0

Fel,w−g−wds = 1

15E∗

√d1s5

F, (38)

Wel−pl =s∫

sF

Fel−pl,w−g−wds

= 1

8πd1pF

[s2-

3

5

(1 + 2

3

(s

sF

) 53)

s2F

], (39)

where sF is the displacement in yield point and s is the currentdisplacement.

If the yield point is very low (sF → 0) in comparison tothe applied force, it can be assumed that plastic deformationstarts from the beginning of loading and the full deformationwork transforms in plastic deformation (WL = Wpl) irre-versibly. The dissipative energy is obtained by integratingEq. (35):

Wpl =s∫

0

Fpl,w−g−wds = 1

2πR1pFs2 = 1

2Fpl,Us

= 1

2kn,pl,w−g−ws2. (40)

For some elastic-perfectly plastic granules the elastic strainenergy is large in comparison with energy of the plastic defor-mation [78]. In this case, the yield point must be considered inthe expression for the plastic force in Eq. (35) as s := s− sF:

Fpl,w−g−w = 1

2πd1pF (s − sF) . (41)

The deformation work of elastic-perfectly plastic granuleWel,pl results from the sum of the elastic strain energy upto the yield point in Eq. (38) and the energy of the plasticdeformation done by the force in Eq. (41). Insertion of theyield pressure (Eq. (26)) into Eq. (41), this deformation workresults in:

Wel,pl =sF∫

0

Fel,w−g−wds +s∫

sF

Fpl,w−g−wds

= 1

5E∗

√d1s5

F

[1 + 15

2

(s

sF− 1

)2]

. (42)

Additionally during granule stressing one has to considerfriction between the primary particles sliding within theirmicroscopic contacts, see Tomas [98,99]. The frictionalmicrocontact behaviour contributes remarkably to the energyabsorption of the stressed granule.

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2.5 Description of energy absorption using a restitutioncoefficient

2.5.1 Measurement of the restitution coefficientof a granule

The restitution coefficient is a very important parameter todescribe the particle collisions in rapid particle or particle-fluid flows [104]. This is an essential parameter which isneeded to describe the energy absorption and damping forcein numerical discrete modelling of particles [50,61,74,86].The restitution coefficient is used in the field of granularmatter and fluidized grains, more known as “granular gas”[22,73], to model the inelastic behaviour of grains; see mod-els of plane flows of circular disks [37], inelastic particlesin a general flow field [62]. Moreover, several hydrody-namic models for granular gases are based on the hard sphereapproaches, which include the restitution coefficient [9]. Thecomplex dynamic evolution of granular gases, where theenergy loss takes place due to inelastic deformation and fric-tion by collisions, can be predicted using the models withnormal and tangential restitution coefficients [114].

Figure 7 shows the contact force versus time and displace-ment in the impact between a comparatively soft sphericalgranule and a fixed stiff wall at different deformation behav-iour. During the initial period of impact, i.e. compression,the contact partners are compressed and their kinetic energyis transformed into internal energy of deformation, frictionand adhesion. The reaction force deforms the particle, whichleads to the contact displacement or overlap. With increas-ing indentation the contact force increases and the relativevelocity of contact partners decreases. If the deformation israte-independent (elastic or plastic behaviour), the maximumindentation and maximum contact force occur at zero relativevelocity [88].

The elastic contribution of the impact energy absorbedduring the compression is released during the restitutionphase of impact and leads to the elastic force that separatesthe contact partners. The absorption of the kinetic energy dur-ing the impact can be described by a restitution coefficient.The coefficient of restitution is a ratio of impulse during therestitution phase of impact (tC ≤ t ≤ tR) to that during thecompression phase (0 ≤ t ≤ tC ), Eq. (43). Moreover, theimpulse ratio in Eq. (43) gives the square root of the ratio ofelastic strain energy Wkin,R released during the restitution tothe impact energy, i.e. initial kinetic energy Wkin .

e =∫ tR

tcFdt∫ tc

0 Fdt=

√Wkin,R

Wkin=

√1 − Wdiss

Wkin

= v1,R

v1≈

√h1,R

h1. (43)

Both contributions of impact energy can be obtained bythe force-displacement curve, Fig. 7b. Hence, the restitutioncoefficient is the square root of the ratio of the area belowthe curve during unloading (restitution) to that during loading(compression).

In case of a fully elastic impact the impact energy adsorbedduring compression is fully restored during the rebound andso the relative velocity of contact partners before impactis equal to that after the impact, e = 1 in Eq. (43). In thecase of full absorption of initial kinetic energy due to plas-tic deformation, adhesion, friction in the contact as well aspropagation of the stress waves, the impact bodies are notseparated after the unloading (restitution), e = 0 in Eq. (43).The force-displacement curve of perfectly plastic contactpartners does not show elastic restitution as illustrated inFig. 7b. For elastic-plastic behaviour during the impact, therestitution coefficient is in the range of 0 < e < 1, see examplesin [5,15,16,20,23,28,34,55,59,60,71,76,83,85,88,91,92].

Two different types of impact, i.e. normal and oblique, aredescribed by normal and tangential restitution coefficients:

en = vn,R/vn, (44)

et = vt,R/vt , (45)

where vn and vt are normal and tangential components of thevelocity vector, respectively.

Different types of equipment were developed to measurethe restitution coefficient, which can be divided into free-fall(a), particle–particle (b) and pendulum tests (c, d) illustratedin Fig. 8.

The most frequently used method to measure the restitu-tion coefficient in both normal and tangential direction is thefree-fall test (Fig. 8a), see examples in [23,60,71,76,85,92].To describe the oblique impact the measurements are carriedout at different impact angles, in the range from 0◦ to 90◦.The additional information about the impact behaviour canbe obtained from rebound angle and the angular speed ofrotation created during the rebound [46,18].

Neglecting the fluid drag force acting upon particle duringthe free fall and rebound, the normal coefficient of restitutioncan also be determined as a ratio of heights after and beforeimpact, h1,R and h1 in Eq. (43). These are measured by asimple free-fall test.

In the publications [21,23,60] the restitution coefficientwas measured without any initial spin of particles. The usingof a vacuum nozzle prevents the initial rotation of the par-ticle by releasing. On the other hand, to give a predefinedrotation before collision the particle can be wrapped with astrip of paper, which will be unwrapped during the fallingunder gravity [18]. The angular velocity of the particle canbe varied using strips with different unwrapped length. Adisadvantage of this method is the possible slip of particleduring the unwrapping.

123


Fig. 7 Schematicrepresentation of the contactforce versus time (a) anddisplacement (b) at differentdeformation behaviour duringthe impact

Fig. 8 Different testingmethods for measurements ofthe restitution coefficient ofparticles

In many cases, the material of the target for experimentsis chosen identical to walls of the apparatus or the tube ofthe investigated process. Using particle tracking velocimetry,Sommerfeld and Huber [85] obtained the influence of wallroughness and stiffness on the impact behaviour of glass andquartz particles in a particle-laden horizontal channel flow.Some authors have performed the impact of particles on thetarget with a layer, Fig. 8d. The knowledge of these impactcharacteristics is necessary to describe particle collisions inthe presence of solid or liquid layers that occurs in some pro-cesses such as wet comminution, fluidized bed spray granu-lation, filtration etc. Kantak and Davis [42] have performedimpact tests of steel and Teflon balls with wet or dry porouslayers placed on a flat target from quartz. In the work ofHuang et al. [32] the effects of milling conditions includingimpact velocity, ball size and powder thickness were stud-ied by fall-tests. During these experiments a steel ball fellonto steel powder particles located in a shallow recess in thehardened steel plate.

For the particle–particle impact experiments, in contrastto the above category of devices, two particles collide with-out any rigid tool according to (Fig. 8b) [15,20,55,59]. Thistype of stressing occurs in fluidized bed granulation and com-minution. The experimental device consists of two vacuumtweezers that are placed on top of each other to release theparticles. The top particle is dropped first and reaches in a

definite time the falling height of the bottom particle. Beforethe impact the bottom vacuum noise of the second particle isquickly removed. A high-speed camera records the collision.To reach the central impact the release mechanisms of thesedevices should be accurate centred.

Pendulum-based experiments are also performed tomeasure the restitution coefficient in particle-wall [16,34],particle–particle [86,108] and particle-beam events [83].Figure 8c schematically shows the pendulum experiments.Two particles are attached with thin wires to a horizontaloverhead plate at a distance l. During the experiment theparticles are simultaneously released and collide in normaldirection. The relative impact velocity of a particle can becalculated from the distance between the overhead plane andimpact point. The velocity of particles after the impact can bemeasured using high-speed video recording of impact [108]or some photodiodes [86]. The advantage of the pendulum-based experiments is the possibility to perform the impact atsmall impact velocities (cm/s), which is difficult to obtain infree-fall experiments because on the small falling height.

2.5.2 Significant factors affecting the restitution coefficient

The coefficient of restitution can depend on many param-eters, like impact velocity, material behaviour of impacted

123


Fig. 9 Coefficient of restitution of spherical indentor made of tung-sten carbide impacted on a PVC target as a function of the carbidetemperature for four different impact velocities [75]

Fig. 10 The force-displacement curves according to the model ofWalton and Braun

bodies, particle size, shape, roughness, moisture content,adhesion properties and process conditions, like temperature.

Increasing the material temperature can change the con-tact behaviour of particles from dominantly elastic to elastic-plastic or viscoplastic. In this case the restitution coefficient isdecreased. Figure 9 shows the results of impact tests betweencarbide and PVC at different temperatures. For a given impactvelocity the decrease of the restitution coefficient with tem-perature increasing from −20◦C to 70◦C (in the glassy state)is relatively small while a severe decrease can be observedat a temperature above ∼70◦C, as PVC begins to behave asa viscoplastic material [75].

As long as at least one of the contact partners does not startto yield, the restitution coefficient remains constant and onlydepends on material properties. If the contact force exceedsthe yield point, the coefficient of restitution can decreasewith increasing impact velocity, which was shown in manyexperimental studies [21,23,46,64,86] and described withthe models [73,94].

This occurs because of increase in the energy absorptionat higher impact velocities due to plastic deformation andheating of bodies contacted. In the case of constant resti-tution coefficient the ratio of the energy absorption duringthe impact to the kinetic impact energy in Eq. (46) remains

constant for each impact velocity. This behaviour describesthe well-know model of Walton and Braun [106] based onsimple linear force-displacement relationships for loading(FB = kL × s) and unloading (FE = kU × (s − sE )), Fig. 10.Thus the ratio of energies from Eq. (43) can be reduced tothe ratio of constant contact stiffness of load kL to contactstiffness kU of unload:

e =√

Wkin,R

Wkin=

√√√√∫ sU

sEkU (s − sE )ds∫ sU

0 kLsds=

√kU (sU − sE )2

kLs2U

=√

kL

kU. (46)

The expression of Walton and Braun [106] is only one exam-ple of an approximation of particle impact behaviour that waspresented here in detail since we used this model in the fol-lowing description of restitution coefficient of the examinedgranules. However, many different models were developedto describe the restitution coefficient of viscoelastic (Schaferet al. [79], Stronge [88], Stoianovici and Harmuzlu [87]),viscoelastic-adhesive (Thornton and Ning), viscoelastic(Brilliantov [11]), elastic-perfectly plastic (Thornton [93],Wu et al. [109]), elastic-perfectly plastic adhesive (Thorn-ton and Ning [94]), elastic-perfectly plastic with harden-ing (Mangwandi et al. [64]), elastic-plastic (Johnson [38]),elastic-viscoplastic (Adams et al. [3]), plastic and viscoplas-tic (Walton and Braun [106]), elastic-adhesive (Thornton andNing [94]) contacts. A review of these models is given inTable 1.

3 Methods and model granules

3.1 Examined granules

Four different industrial spherical granules—γ-Al2O3, syn-thetic zeolite 13X, synthetic zeolite 4A and sodium benzoate(C6H5CO2Na) were used as test materials (Fig. 11).

Previously we have tested many other granules to findsome model materials and selected these four granulesfor varying the deformation behaviour from elastic-plastic(γ-Al2O3, zeolites) to plastic (sodium benzoate). An advan-tage of these particles is their comparable sizes. Because of anearly ideally spherical shape (sphericity in the range of 0.91-0.95) it is possible to apply the contact theory of spheres fordescription of deformation and breakage behaviour of thesegranules.

A summary of the material properties of these granules isgiven in Table 2. The methods used to measure the propertieswere described in a previous publication [6]. We used twofractions of sodium benzoate in order to determine the influ-ence of the particle size on the stiffness and restitution coef-

123


Tabl

e1

Mod

els

for

cont

actf

orce

low

and

coef

ficie

ntof

rest

itutio

nin

norm

aldi

rect

ion

Aut

hor

Beh

avio

urFo

rce-

disp

lace

men

tlaw

Mod

elof

rest

itutio

nco

effic

ient

eIn

dex,

rem

arks

Her

tz[3

0]Id

eally

elas

ticF l

=F

u=

2 3E

∗√ R∗ s

3e

=1

E∗ e

ffec

tive

mod

ulus

ofel

astic

ity,

Eq.

(4)

R∗

med

ian

part

icle

radi

usin

Eq.

(5),

spa

rtic

lece

ntre

appr

oach

orto

tal

over

lap

heig

ht

Stro

nge

[88]

Vis

coel

astic

,e

=co

nsta

ntF

n=

Fu

=k e

lsel

=η

s∗,

s=

s el+

s de

=ex

p

( −π

√ 4η2/k e

lm∗ −

1

) ,

η>

1 2

√ m∗ k

el

*M

axw

ellm

odel

ηda

mpi

ngco

effic

ient

,k e

lspr

ing

cons

tant

,s e

lsp

ring

disp

lace

men

t,s d

dash

potd

ispl

acem

ent,

m∗

effe

ctiv

em

ass

Schä

fer

etal

.[79

]L

inea

rvi

scoe

last

ic,

e=

cons

tant

F l=

Fu

=k e

lsel

+η

s∗,

s=

s el=

s de

=ex

p

( −π

η√ 4m

∗ kel

−η2

)*

Kel

vin-

Voi

gtm

odel

Stoi

anov

ici&

Har

muz

lu[8

7]N

on-l

inea

rvi

scoe

last

ic,

vari

able

eF l

=F

u=

k els

+η|s|

s∗,

s=

s el+

s de

≈( η k e

lv

+ex

p( −0

.4η k e

lv)) −

1*

Kel

vin-

Voi

gtm

odel

vim

pact

velo

city

Bri

llian

tov

etal

.[11

,72]

Non

-lin

ear

visc

oela

stic

,va

riab

lee

F l=

2 3E

1−v

2

√ R∗

( s3/2+

3 2A

d√ ss

)e

=1

−b

( v m∗2

) 1/5

Ad

=1 3

(3η

2−η

1)2

(3η

2+2

η1)

( 1−v

2) (1

−2v)

Ev

2

b=

1.15

( 3A

d2

)(2 3

E∗√

R∗) 2/

5

E=

E1

=E

2,v

=v

1=

v2

Wal

ton

&B

raun

[106

]V

isco

plas

tic,

vari

able

eF l

=k l

s,F

u=

k u(s

−s u

),

k u=

k l+

SF

u,m

ax

e=

√ k l k u=

√k l

k l+S

Fu,m

axF

u,m

axm

axim

umun

load

ing

forc

e

=( 1

+Sv

√ 2kl/

m

) −1/2

Tho

rnto

n[9

3]E

last

ic-p

erfe

ctly

plas

tic,

non-

adhe

sive

Fel

,l=

4 3E

∗ l

√ R∗ s

3fo

rs

≤s F

,e

=√ 2.

08

( 1−

v∗2 F 6

)(v∗ F

v∗ F+2

√ 6 5−

1 5v∗

2 F

) 1/4

for

v≥

v F,

E∗ l,

E∗ uef

fect

ive

You

ng’s

mod

uli

duri

ngth

elo

adin

gan

dun

load

ing:

Fpl

,l=

FF

+π

pF

R∗ (

s−

s F)

v∗ F=

s F v,v

F=

( πpF

2E

∗) 2√ 2 5

pF ρ

E∗

=( 1−

v2 1

E1

+1−

v2 2

E2

) −1

Fu

=4 3

E∗ u

√ R∗ (

s−

s u)3

for

s≥

s F,

vim

pact

velo

city

,

v Fyi

eld

velo

city

FF

forc

ein

the

yiel

dpo

int

Fel

,l(s

F)

123


Tabl

e1

cont

inue

d

Aut

hor

Beh

avio

urFo

rce-

disp

lace

men

tlaw

Mod

elof

rest

itutio

nco

effic

ient

eIn

dex,

rem

arks

Tho

rnto

n&

Nin

g[9

4],

John

son

&Po

llock

[39]

Ela

stic

-per

fect

lypl

astic

,adh

esiv

ee

=0

for

v≤

v sv s

stic

king

velo

city

e=

[ 1−

( v s v) 2] 1/

2fo

rv s

≤v

≤v F

v s=

1.84

( 4 (/

R1)5

ρ3

E∗2

) 1/6

e=

[ 2.08

( 1−

v∗2 F 6

)(

v∗ F

v∗ F+2

√ 6 5−

1 5v∗

2 F

) −( v s v

) 2] 1/2

for

v≥

v F

in

terf

ace

ener

gy

John

son

[2]

Ela

stic

-pla

stic

,no

n-ad

hesi

veE

last

ic:a

cc.H

ertz

ρpa

rtic

lede

nsity

Ela

stic

-pla

stic

:F l F

F=

0.01

1

( F lE

∗2R

∗2σ

3 F

) ,s s F

=0.

037

( sE∗2

R∗2

σ2 F

)e

=1.

72

(σ

5 FE

∗4ρ

) 1/8

v−1/

4s

=( 27

πF

σF

4E

∗2) 1/

2

Fully

plas

tic:,

F l FF

=0.

2

( sE∗2

R∗2

σ2 F

) =5.

5s s F

,Fo

rfu

llypl

astic

impa

ct

Unl

oad:

.Fu

FF

=5.

06·1

0−4( sE

∗2R

∗ σ2 F

) =0.

38( s s F

) .

Ada

ms

etal

.[3]

Ela

stic

-vis

copl

astic

,no

n-ad

hesi

veF

n,l=

3πdσ

′ Fs,

σ′ F

=k′

¯ εq=

k′v d,

Fn,

u=

1 2

( r k,m

axσ

FE

∗3(s

−s E

)3) 1/

2,

s E≈

s max

−7.

1r k

,max

σF

E

e=

√ 18π

/5

(σ

2 FE

∗ σ′ F

) 1/2( 9σ

′ F8ρ

) 1/8

v−1/

4

Atl

owim

pact

velo

citie

s:

σ′ F

≈σ

F,e

≈3.

4

( σ5 F

ρE

∗4

) 1/8

v−1/

4

dpa

rtic

ledi

amet

er,

σ′ F

cons

tant

valu

eof

flow

stre

ss,

k′ap

pare

ntpl

astic

flow

[2],

σF

unia

xial

visc

opla

stic

stra

in,

r k,m

axan

ds m

axco

ntac

tra

dius

and

disp

lace

men

tatm

axim

umfo

rce,

s Epe

rman

ent

disp

lace

men

taf

ter

unlo

adin

gE

∗ as

inth

em

odel

ofT

horn

ton

[93]

Wu

etal

.[10

9]E

last

ic-p

erfe

ctly

plas

ticA

ppro

xim

atio

nby

FEM

e=

n[(

v v F

) (σ

F E∗)] −

1/2

n=

0.78

for

impa

ctel

astic

sphe

rew

ithan

elas

tic-p

erfe

ctly

elas

ticha

lf-s

pace

123


Tabl

e1

cont

inue

d

Aut

hor

Beh

avio

urFo

rce-

disp

lace

men

tlaw

Mod

elof

rest

itutio

nco

effic

ient

eIn

dex,

rem

arks

n=

0.72

for

impa

ctel

astic

sphe

rew

ithan

elas

tic-p

erfe

ctly

plas

ticha

lf-s

pace

,n

=0.

58fo

rim

pact

elas

tic-p

erfe

ctly

plas

ticsp

here

with

ari

gid

wal

lE

∗as

inth

em

odel

ofT

horn

ton

[93]

Man

gwan

diet

al.[

64]

Ela

stic

perf

ectly

plas

tic,

with

stra

inha

rden

ing

Fn,

l=

πr2 k

H=

πr2 k

[ z

( r2 k R∗) n]

e=

3π√ 3 10

[ σ2 F E∗ u

(x+

y )3/

(2n+

4)] −1

/2

×[ 15

E∗ l

16R

∗2(x

+y )

5/(2

n+4 )

+kπ

(2+n

)R

∗n+1

y] −1/2

x=

( 9πR

∗ σY

4E

∗ l

) 2n+4

,y

≈(n

+2)( m

v2) R

∗n+1

kπ

Hha

rdne

ss,

r kco

ntac

trad

ius,

zsi

zede

pend

entc

oeffi

cien

t,n

hard

enin

gco

effic

ient

,E

∗ lan

dE

∗ uE

-mod

uli

duri

nglo

ad-

ing

and

unlo

adin

g,ac

c.to

Tho

rnto

n[9

3]

123


Fig. 11 Digital images of examined granules

Table 2 Characteristics of the examined granules

Characteristics γ-Al2O3 Zeolite 4A Zeolite 13X Sodium benzoate

Chemical composition in % 97.9% γ-Al2O3 85% Syntheticzeolite 4Abinder: clay

85% Syntheticzeolite 13Xbinder: clay

C6H5CO2Na

Granule size in mm 1.6–1.9 2.1–2.5 1.2–1.7 0.86–0.96, 1.24–1.60

Sphericity 0.95 0.91 0.94 0.92

Granule density in kg/m3 1040 1140 1150 1440

Solid density in kg/m3 3230 3640 2100 1500

Specific surface area in m2/g 145 159 415 7.6

Pore volume in % 68 69 45 4

Water content in kg water/kg solid 0.023 0.002 0.006 0.007

Application Drying processes Adsorbent (molecular sieve) for dryingprocesses and cleaning of gas

Food and beverage preservative

Manufacturer Sasol, Hamburg CWK Chemiewerk Bad Köstritz, BadKöstritz

DSM, Geleen

ficient. The compression and free-fall tests were performedat ambient temperature of 25◦C and atmospheric pressure.

3.2 Compression tests

A compression test of a single particle up to primary break-age is used to determine the minimum energy requirementfor breakage, as shown first by Carey and Bosanquet [14].The absorption of stressing energy in particle bed crushing ismuch higher than in compression of a single particle becauseof friction between the particle contacts. In a compression testthe secondary breakages (after the primary breakage) can beseparately observed, due to the comparatively low deforma-tion velocity (from mm/ min to cm/ min).

In the pioneering works of Rumpf [77] and Schönert [80],the deformation and breakage behaviour of solid particleswas described by force-displacement curves measured bycompression tests. Recently, the granules have been inves-tigated using compression test: Al2O3 produced by fluid-ized bed-spray granulation [4], detergents [78], enzyme [10],polymer bound Al2O3 granules [84] and calcium carbonate[64].

Fig. 12 Principle of single particle compression test

Figure 12 shows the principle of a granulate compressiontest. During the movement of the punch towards the upperfixed plate, a contact between the particle and the fixed plate iscreated. During the stressing period, displacement and forcevalues are measured. Compression tests are carried out eitherwith strain control (constant stressing velocities) or stresscontrol (fixed stress or load rate).

123


Because of fatigue of solids during the cyclic loadingtheir breakage can occur at stresses that are substantiallylower than the failure stress during static loading. The reduc-tion of fracture strength occurs because of formation andpropagation of shear zones and microcracks during eachcycle. The number of cycles up to fracture decreases withincreasing stress amplitude, which was already describedfor metallic and ceramic materials with the Wöhler curve[76]. The investigations of solid particles and agglomeratesdemonstrate the considerable effect of repeated loading onthe mechanical behaviour and breakage point. Tavares andKing [92] explained a decrease of the elastic-plastic stiff-ness of particles during repeated impacts as well as theirbreakage because of the formation and propagation of dam-ages. In opposition to this, the stiffness of spherical granulesincreased with increasing number of loading/unloading com-pression cycles up to the point that saturation of the plasticdeformation is reached [5]. The intensity and the frequency ofstressing, the particle size and the microstructure influencematerial resistance against cyclic loading. Beekman et al.[10] and Pitchumani et al. [71] have also confirmed theseeffects for different granules.

During a repeated compression test, the punch movestowards the upper plate and presses the granule up to a definedforce. Subsequently, the punch moves downwards, thus theunloading of the granule takes place.

The compression tests of the model granules performed inthis work were repeated 100 times for each sample at a con-stant stressing velocity of 0.02 mm/s to increase the statisticalsignificance of results.

3.3 Apparatus and methods for free-fall experiments

Figure 13 shows the experimental setup for the free-fallexperiments. Before it is dropped, a granule is held at a heighth1 above the target (flat plane) with the aid of vacuum twee-zers. The granule falls freely onto a target and reaches arebound height h1,R after the impact. The movement of thegranule near the contact point before and after the impactis recorded by a high-speed video camera with a frequencyof 4.000 frames per second and a resolution of 512 × 256pixels.

From the ratio of energies in Eq. (43), it follows that thecoefficient of restitution is a ratio of relative rebound velocityv1,R to that before impact v1. These velocities were deter-mined from captured impacts in free-fall experiments withhelp of self programmed software on the basis of Matlab.

In order to examine the influence of the impact velocityon the restitution coefficient of granules, the drop height wasvaried from 12 mm to 1.5 m, i.e. the impact velocity wasin the range of about 0.5–4.5 m/s. This range was selectedaccording to two criteria. The upper limit had to be belowthe critical impact velocity that leads to the breakage of gran-

ules. In previous work [6] a critical velocity of 6.1 m/s forzeolite 13X was found through impact tests. The performedmeasurements showed that the standard deviations of theimpact and rebound velocities increase with increasing in thefall height. On the other hand, at heights below 12 mm theerror of measurement greatly increases. Therefore the lowerlimit of the used impact velocities corresponds to the reboundvelocitywhich can still be measured accurately.

During the free-fall tests, the angle of incidence � was var-ied from 0◦ to 80◦. To describe the oblique impact the angle�R and angular velocity ω1,R of granules after rebound fromthe wall have been measured. To detect the relatively smallgranules, a black background was installed.

For accurate measurement of the normal restitution coeffi-cient normal impact and rebound should be ensured. Duringour experiments some granules without initial spin fallingat normal impact rebounded of the horizontal plate underan angle (0◦–10◦). This can be explained by the non-idealspherical shape of the granules. Furthermore, the reboundbehaviour of granules can be affected by defects of particlesurface [6] when they occur in the contact range. However,during data processing no influence on the mean restitutioncoefficient was observed for rebound angles below 8◦. There-fore, we determined the coefficient of restitution using free-fall tests, in which the rebound took place in this small anglerange. The free-fall test was repeated for each model materialuntil 50 nearly vertically rebounds (angle below 8◦) were cap-tured. The number of valid measurements was about 80% forγ-Al2O3, 30% for zeolites 13X and 4A and 20% for sodiumbenzoate.

Neglecting the air resistance acting on the granule therestitution coefficient can be calculated as the ratio of dropand rebound heights in Eq. (43). However, provided that theheight is large and the granule has a small density, the airresistance decreases the velocity of the particle greatly, whichleads to a significant error in the restitution coefficient cal-culated through the heights.

The real velocity v1 of a free falling granule can be calcu-lated depending on granule and air properties from the fol-lowing force balance or equation of motion (also see Fig. 13):

m1dv1

dt= n

(Fg − FB

) − FD, n ={

1 f all−1 rebound

, (47)

where m1 is the mass of granule, Fg is the gravity force, FB

is buoyancy force and FD is drag force.For an ideally spherical granule with diameter d1, and

density ρ1 Eq. (47) becomes:

m1dv1

dt= n

πd31

6g (ρ1 − ρf) − cD

πd21

4

ρf v21

2, (48)

where cD is the drag coefficient and ρ f is air density.The drag coefficient can be calculated according to Kaskas

[43], which is applicable over the entire range of Reynolds

123


Fig. 13 The schematicrepresentation of free-fall device

numbers (0 < Re < 2-4 ×105) [81]:

cD = 24

Re+ 4√

Re+ 0.4, (49)

where the particle Reynolds-number Re is given by:

Re = v1d1ρf

ηf, (50)

where η f is dynamic viscosity of air.The impact and rebound velocities of the examined gran-

ules were obtained by numerical solution of Eq. (48) using thedrag coefficient from Eq. (49) (at η f = 1.8×10−5kg/ms andρ f = 1.19kg/m3) and the material properties from Table 2.An example of the results is shown in Fig. 14 for γ-Al2O3

granules. A noticeable difference between the velocities cal-culated with and without air resistance and buoyancy forcescan be observed at the heights above 0.2 m. Hence, the largeparticle deceleration during fall and rebound due to mainly airresistance (the buoyancy force can be neglected with respectto the gravity, ρ f << ρ1) confirmed that the drag cannot beneglected in this case. In our free-fall experiments mentionedbelow, the real velocities of the granules were observed byhigh-video recording of the impacts. The measured results(included in Fig. 14 for γ-Al2O3) confirmed the calculationfor all examined granules.

Another point that must be considered to accurately mea-sure the restitution coefficient is the energy absorption due tostress waves. Elastic stress waves arisen in the contact areacan have significant effects on the restitution coefficient [48].At the beginning of contact deformation, the spherical elasticwave expands away from the contact region into the target.After the reflection from target borders, the wave comes back

Fig. 14 Comparison between measurement and calculation of impactand rebound velocities at different drop heights for γ-Al2O3 granules(d1 = 1.8 mm in calculation and d1 = 1.6 − 1.9 mm in experiments)

towards the contact area. If the contact time is longer than theperiod of wave propagation through the plate from contactpoint to the bottom wall of plate and return to the contactpoint then the waves will reach the contact partners and leadto loss of kinetic energy which was first calculated by Hunter[33]. Because of reflected waves the granule looses additionalenergy which leads to a decrease of the restitution coefficient.Koller [48] estimated this energy loss to about 3.6% for a steelball impacted on a glass plate with the thickness of 19 mm.

Thus, the thickness δ, width B and length L of the usedtarget must be sufficiently large to exclude the energy absorp-tion due the reflected elastic waves:

δ >vl × tR

2, B, L ≥ vt × tel

2. (51)

123


The propagation speed of longitudinal (vl) and transversal(vt ) waves through the target can be calculated from modu-lus of elasticity E2, Poisson’s ratio ν2 und density ρ2 of thetarget [56]:

vl =[

E2 (1 − v2)

ρ2 (1 + v2) (1 − 2v2)

]1/2

, vt =[

E2

2ρ2 (1 + v2)

]1/2

=[

G2

ρ2

]1/2

. (52)

As target during the free-fall experiments a hardened steelplate (E2 = 2.1×105N/mm2, ν2 = 0.25, ρ2 = 7, 850 kg/m3)

with a smooth surface was used. In this material the longitu-dinal waves propagate at a velocity of vl = 5, 367 m/s anda transversal wave velocity of vl = 3, 626 m/s. The time ofimpact tR in Eq. (51) can be estimated using Eq. (17) takinginto account the properties of the granules mentioned below(see Tables 2 and 3). Finally, the minimal thickness of thetarget was found to be 21 mm. Since the behaviour of gran-ules is not ideally elastic, the time of contact is higher thancalculated using Hertz theory. Therefore, for the experimentsa target thickness of 30 mm was employed. The width andlength of the plate (210 mm) are less important, since thetransversal wave with vt < 0.64 × vl is relatively slow.

4 Experimental results

4.1 Material parameters

The typical force-displacement curve for γ-Al2O3 granulesis shown in Fig. 15. At the beginning of the punch-parti-cle contact the elastic contact deformation of the granuletakes place. The elastic force was described by Hertz theory,Eq. (10). The modulus of elasticity and stiffness of the gran-ule during elastic deformation were calculated from Eqs. (4)and (11), respectively. The average values of the mechani-cal properties of the examined granules are summarised inTable 3.

Due to the initial parabolic curvature of F(s), the contactstiffness increases with increasing displacement and reachesthe maximum value in the yield point F in Fig. 15. In thispoint the plastic deformation begins. This is confirmed bythe increasing deviation of the experimental curve F-B fromthe theoretical Hertz curve (Fig. 15 right).

The slope of the curve F-B is a measure of the elastic-plastic stiffness, which is proportional to the granule radiusand the yield strength, according to Eq. (30). A small slopeof the curve implies more plastic behaviour, whereas a largeslope indicates “stiff” material behaviour. In Eq. (26), theyield strength pF can be characterized as the beginning of theplastic displacement sF at contact force FF . These param-eters were determined from the force-displacement curves Ta

ble

3M

echa

nica

lcha

ract

eris

tics

ofth

eex

amin

edgr

anul

esby

com

pres

sion

Gra

nule

sD

iam

eter

Bre

akag

efo

rce

Yie

ldpo

int

Mod

ulus

ofel

astic

itySt

iffn

ess

inno

rmal

dire

ctio

nin

N/m

mC

onta

ctar

eaco

effic

ient

d 1in

mm

FB

inN

FF

inN

s Fin

µm

E1

inG

Pak n

,el,

gat

F=

FF

k n,e

l−pl

,gat

F=

0.98

FB

k n,p

l,g

κA

ats

=0.

98s B

γ-A

l 2O

31.

6–1.

939

.0±

3.6

8.5

±2.

012

.5±

3.8

14.5

±0.

3121

20±

330

1819

±17

4–

0.79

Zeo

lite

4A2.

1–2.

526

.1±

4.4

16.4

±0.

950

±5.

02.

7±

0.21

983

±76

893

±49

–0.

70

Zeo

lite

13X

1.2–

1.7

8.8

±1.

11.

0±

0.08

10±

0.9

2.3

±0.

1629

4±

3425

6±

26–

0.81

Sodi

umbe

nzoa

te0.

80–0

.96

6.2

±2.

1<

0.1

<1

––

–28

8±

611.

0

1.24

–1.6

016

.3±

2.9

<0.

1<

1–

––

491

±98

1.0

123


Fig. 15 Typicalforce-displacement curve ofγ Al2O3 granules(d1 = 1.62 − 1.76 mm) duringcompression (vB = 0.02 mm/s)

Fig. 16 Comparison of calculated and measured contact stiffness ver-sus displacement of a typical zeolite 13X granule (d1 = 1.4 mm) atcompressive stress (vB = 0.02 mm/s)

(Table 3) and thereby the force in elastic-plastic deformationrange was approximated. The small deviation of the theoreti-cal curve from experimental data is explained by the fact thatthe contact area of the granule under compression was notperfectly circular and had visible roughness (Fig. 18) [5].

The contact stiffness of a granule during elastic-plas-tic deformation (listed in Table 3) was calculated usingEq. (30) taking into consideration that kn,el−pl,g = 2 ×kn,el−pl,w−g−w. The force-displacement curve changes in the

yield point from the elastic to the elastic-plastic range. There-fore, the contact stiffness drops at this point. Fig. 16 showsthe development of the stiffness during the full deformationrange of a zeolite 13X granule. After a severe decrease inyield point the stiffness gradually decreases during the elas-tic-plastic deformation up to the breakage point where thestiffness suddenly drops. It is noted here that the elastic-plastic stiffness of γ Al2O3 and both zeolite granules exceedstheir peak values at the yield point.

Fig. 17 Typical force-displacement curves of sodium benzoate gran-ules during compression (vB = 0.02 mm/s)

Using the elastic-plastic contact area coefficient κA inTable 3 the amount of plastic deformed contact area Apl incomparison to the total contact area can be observed. Close tothe breakage point the coefficient κA of γ-Al2O3 and zeolite13X reaches the value of ∼4/5. According to Eq. (22) thisresult corresponds to a contact area ratio of Apl/AK = 2/5.

The breakage of the γ-Al2O3granule follows at a force(point B in Fig. 15), which is approximately three timeshigher than the yield force (point F). Compared to this,the zeolite 13X granule shows a small elastic deformation(Fig. 16), but considerable elastic-plastic deformation beforeprimary breakage.

The force-displacement curve of sodium benzoate is astraight line along the whole deformation region up to thebreakage (Fig. 17). According to Eq. (35), that means thatperfectly plastic deformation occurs when Ael = 0 andAK = Apl . Figure 17 shows the effect of the granule size onthe force-displacement behaviour. In the case of bigger plas-tic granules, both the breakage force and the contact stiffnessincrease. The stiffness of zeolite granules in elastic and elas-

123


Table 4 The contact radius and yield pressure of sodium benzoate granules by compression

Granule diameter d1,50in µm sB in µm dk,pl (at spl = 40µm) in µm rk,pl at sB in µm pF in MPa σmaxin MPa

Eq. (34) Experiment

870 ± 61 43 ± 6 264 245 ± 14 136 ± 14 114 ± 20 10.5 ± 1.8

1390 ± 90 66 ± 8 333 308 ± 18 215 ± 20 112 ± 21 10.7 ± 1.7

Fig. 18 SEM of flattened surface of a sodium benzoate granule afterthe compression test: 1 the entire approximated circular contact area, 2the real contact area, 3 less deformed part of a contact surface (rough-ness depressions)

tic-plastic ranges increases with increasing granule diameterand therefore the material becomes stiffer, see Eqs. (11) and(30). The increase in the breakage force does not influencethe material strength of plastic granules, σmax = FB/

(π R2

1

).

As an example—for different size fractions of sodium ben-zoate (see Table 4) the average compressive strength is about10.6 MPa. Eqs. (34) and (35) can be used to develop theequation for the strength of dominantly plastic granules:

σmax = sB

R1pF ≈ 0.1pF , (53)

where sB is the displacement up to the primary breakagepoint. The constant ratio sB/R1 was experimentally found tobe about 0.1.

At the same deformation value, the formed contact areaincreases with increasing particle size. Figure 18 shows Scan-ning Electron Microscope (SEM) images of a sodium ben-zoate granule after the compression. The flattened contactsurface (1) is not perfectly circular, since the granules sur-face is not evenly curved and contains roughness. Load andcontact plastic deformation occur only at the asperities (2) ofthe contact. Therefore a small part of the area is not deformed(3).

The diameter of the contact dk,pl in Table 4 is calculatedby Eq. (34) at a deformation value of 40µm. This value is ingood agreement with the equivalent contact area measuredusing a microscope. Each average value here correspondsto 10 SEM measurements. We have observed that the yieldpressure pF is independent of the particle size.

4.2 Loading-unloading behaviour of granules

Typical loading-unloading curves of model granules areshown in Fig. 19. Loading and unloading were performedat a constant stressing velocity of 0.02 mm/s. To examine theeffect of the stress intensity on the energy absorption, themaximum loading force was varied. After the first loading-unloading cycle (up to force in point U1 in Fig. 19) the gran-ule was carefully rotated on the punch, to load another rangeon the granule surface during the second loading-unloadingcycle (up to a force in point U2). The magnitude of the sec-ond maximum force was chose to be in the range of 90–95%of the average breakage force of corresponding granules.

After the complete unloading (points Ei ) all granulesshowed a permanent deformation. Furthermore, during thetime of 10 seconds a contribution of this deformation about5% disappeared due to creeping

(Ei → E ′

i

). Note that the

unloading curve of elastic-plastic zeolite and γ Al2O3 gran-ules differs from the loading in the elastic range (Hertz curve)because of softening during the unloading. Therefore, theslope of the unloading curve, or in other words, the stiffnessduring unloading is evidently larger then that during the elas-tic loading. The nearly vertical unloading curves of sodiumbenzoate without elastic deformation prove the dominantlyplastic behaviour of these granules during the compression.

The work done during the compression of the examinedgranules was calculated according to the above derived rela-tions (38–40). The energy absorption due to viscoelastic andviscoplastic deformations is neglected in this calculation. Thegood agreement of the predicted values with experimentaldata can be obtained from Fig. 19 and Table 5.

4.3 Effect of cycle number on the dissipative behaviourat a constant load amplitude

Zeolite 13X and sodium benzoate granules were investigatedby one-way cyclic compression tests. During the measure-ment, a granule was repeatedly loaded and unloaded witha constant velocity of 0.02 mm/s up to constant maximumforce, which is called the load amplitude. This force was setnear the average breakage point of the respective granule.

Figure 20 shows the force-time curves for the first fourcycles. The load amplitude (Fcyc = Fmax), the mean force

123


Fig. 19 Typical loading-unloading curves at different maximum compressive forces. a γ Al2O3,d1 = 1.67 mm; b zeolite 4A, d1 = 2.10 mm; czeolite 13X, d1 = 1.56 mm and d sodium benzoate, d1 = 1.60 mm

Table 5 Energy characteristics of granule compression

Energy in µJ γ-Al2O3at FU = 20N Zeolite 4A at FU =20 N Zeolite 13X at FU = 7.3N Sodium benzoateat FU =11.5 N

Exp. Theor. Exp. Theor. Exp. Theor. Exp. Theor.

Wel,el 44 48 329 345.2 5.2 4.8 – –

Wel−pl 311 310 162 172.9 111.9 112.7 – –

Wpl – – – – – – 203 200

WL 355 358 491 518.1 117.1 117.5 203 200

(Fm) and the period (T) characterize the cyclic stressing pro-cess. As the number of cycles increases, the maximum forceFcyc is reached faster. Thus, the cycle time decreases. Aftereach unloading the testing device needed a time of 10 s for thereverse (pause tu in Fig. 20). The soft reversal of the equip-ment for the unloading leads to creep which can be seen aftereach peak in Fig. 20.

Figure 21 shows the results of a measurement by 20 cyclesfor a zeolite 13X granule. For clarity cycles from 3 to 18were not included in this diagram. At the beginning of eachcycle the granule behaves elastic only until the yield point F.If the loading curve of the second cycle (E1U2) is moved tothe approximated Hertz curve of the first cycle (E1 → E ′

1inFig. 21), it becomes clear that the second loading can be

123


Fig. 20 Force-time curves during one-way cyclic loading in compres-sion of a zeolite 13X granule (d1 = 1.54 mm, Fcyc = 7N, vB =0.02 mm/s)

described by the Hertz equation of the first compression.Therefore, during each following loading the force-displace-ment response in elastic range depends on the modulus ofthe elasticity and on the amount of total plastic deforma-tion accumulated in the previous cycles. The modulus of theelasticity remains nearly constant during the cyclic loading.Therefore it can already be determined during the first cycle.Considering these measurement results, the elastic force incycle z can be described with the force of the previous cycle

(z-1):

Fel,z = Fel,z(sel,z + spl,z−1

) − Fel,z−1(spl,z−1

)

= 1

6E∗√d1

(√(sel,z + spl,z−1

)3 −√

s3pl,z−1

).

(54)

The first derivative of the force-displacement function givesthe stiffness of elastic deformation in cycle z:

k∗n,el,w−g−w,z = dFel,z

ds= 1

4E∗

√d1

(sel,z + spl,z−1

). (55)

The loading curves obtained from Fig. 21 are shown in Fig. 22in transformed coordinates according to Hertz equation (10),where sz is the displacement during a cycle z. The con-tact stiffness in the elastic and elastic-plastic deformationranges increases with increasing number of cycles. The larg-est increase in the loading stiffness occurs by change fromthe first to the second cycle. A further increase of the curveslope can only be observed during the first 15 cycles. There-fore, all loading curves for z > 15 are located on the samecurve O − Uz , which approaches the corresponding Hertzcurve practically up to the maximum force (point Uz). Thehysteresis loops detected between unloading and reloadingcurves are gradually decreased with the progress of load-ing cycles. After some cycles no plastic deformation arisesand the unloading curves return to the loading origin, i.e.a saturation state is reached. However, small viscoelasticdeformations occur during the cyclic loading in the satura-tion state. As a result, the area of the hysteresis loop does not

Fig. 21 Typicalloading-unloading curves duringone-way cyclic loading incompression of zeolite 13Xgranule (d1 = 1.68 mm,

Fcyc = 7N, vB = 0.02mm/s)

123


Fig. 22 Effect of number ofloading-unloading cycles oncontact stiffness of zeolite 13Xgranule (d1 = 1.68 mm, Fcyc =7N, vB = 0.02 mm/s)

fully disappear; see the loop between the loading-reloadingcurves of the 19th and 20th cycles in Fig. 21.

From the curves F(sz) in Fig. 23 it can be observed thatafter 25 cycles the sodium benzoate granule deforms onlyviscoelastic without permanent deformation and so it exhib-its the same loading-unloading response in all further cycles.Thus, the behaviour of both examined granules during cyclicloading changes from respectively elastic-plastic (by zeolite13X) and dominantly plastic (by sodium benzoate) to visco-elastic.

Figure 24 illustrates the development of elastic and plasticstrains during the cyclic loading-unloading for the two exam-ined granules. The increase of cycle number results in gradualdecreasing theplastic strain toazerovalue,whichclearly indi-cates a cyclic hardening of granules and reaching a saturationstate, at which the elastic behaviour becomes dominant. Themaximum elastic strain remains constant at each cycle.

The cyclic hardening can be explained on the basis ofmicrostructure change within the contact of the granule,since the highest compressive stresses are generated there.During the plastic deformation of the granule the structureis greatly deformed within the contact area (see Fig. 25).The largest changes occur at the first cycle, which is alsoshown in increasing slope of the loading curve from thefirst to the second cycle in Fig. 22 and 23. This micro-structure change takes place locally close to contact zone,which was confirmed with the following experiment. Afterthe first loading cycle we turned over (90◦) the granuleto stress two other contacts on the granule surface in thenext cycle. The force-displacement curve measured in thesecond cycle exhibits the same slope as in the first cycle(Fig. 19).

The radius of plastic deformed contact area after unload-ing is the initial contact radius for the reloading. The totaldisplacement and the contact radius in the cycle z can becalculated according to:

sz = spl,z−1 + sel,z + spl,z, rk,z = rk,pl,z−1 + rk,el,z(sel,z

)+rk,pl,z

(spl,z

), (56)

where sel,z is the elastic displacement at cycle z(sel,z lies in arange from 0 to the yield point sF ).spl,z is the plastic displace-ment in cycle z.spl,z−1 is the total plastic contact displace-ment in previous cycles (1 . . . z − 1) , spl,z−1 = ∑z−1

i=1 spl,i ,.Thereby, spl,z−1 characterises the history of the contact dis-placement.

The plastic displacement in cycle z can be described fora cyclically hardening material with the following relation-ship:

spl,z = spl,1/zα, (57)

where spl,1 is the plastic displacement in the first cycle. Theexponent α represents a saturation parameter, which dependson the material properties and the cyclic force amplitude.Higher values of α correspond to the smaller number ofcycles that are needed to approach the saturation state. Thetotal plastic displacement in both contacts of compressedgranule during z cycles accumulates from the plastic dis-placements of individual cycles:

spl,tot,z =z∑

i=1

spl,i . (58)

From the approximation according to Eq. (57) (Fig. 26) thefollowing values of the saturation parameter were obtained:

123


Fig. 23 Typicalloading-unloading curves duringone-way cyclic loading incompression of sodium benzoategranule (d1 = 1.76 mm, Fcyc =16N, vB = 0.02mm/s)

Fig. 24 The elastic and plasticcompressive strains versus thenumber of cycles: a zeolite 13X,d1 = 1.68 mm, Fcyc = 7N,b sodium benzoate,d1 = 1.76 mm, Fcyc = 16N

Fig. 25 SEM of surfacemicrostructure of a sodiumbenzoate granule before (a) andafter (b) compression (contactarea). One can see locallydeformed primary particles,shear and frictional slidingbetween the contacts of theprimary particles and formationof shear bands (1). This leads toenergy absorption by friction

α = 2.0 for zeolite granules and α = 1.66 for sodium benzo-ate granules. In addition to Fig. 24 the curves in Fig. 26 showthe difference in the plastic displacement of both granules.

Combining Eqs. (56) and (57), the total displacement ofthe granule in a cycle z can be written:

sz = sel,z +z∑

i=1

spl,1

iα. (59)

Substituting Eq. (59) into Eq. (11), we obtained the stiffnessduring the elastic displacement sel,z at each cycle for morethan two cycles:

kn,el,w−g−w,z = 1

4E∗

√√√√d1

(sel,z +

z∑i=2

(spl,1

(i − 1)α

)),

by z ≥ 2,i ∈ [2, z] . (60)

123


Fig. 26 The total plastic displacement of granules versus the numberof one-way loading-unloading cycles (sodium benzoate: d1 = 1.76 mmat Fcyc = 16N, vB = 0.02mm/s and zeolite 13X: d1 = 1.68 mm atFcyc = 7N; vB = 0.02 mm/s)

Two parameters (the plastic displacement in the first cyclespl,1 and saturation parameter α) are needed to calculatethe elastic stiffness in each cycle. The development of thecontact stiffness during loading, unloading and reloading fora zeolite granule is plotted in Fig. 27. Due to increase ofcontact area the contact stiffness increases, see Eq. (8). Theincrease of the stiffness from the beginning of the first cycleto the yield point (O1 − F1) was calculated using Eq. (11)for elastic stiffness. In the same way, the above developedelastic-plastic model in Eq. (30) was used to approximatethe stiffness after the beginning of the plastic deformationup to the point in which unloading starts (F ′

1 − U1). Atthe beginning of unloading (U ′

1) the stiffness is well higherthan during loading. The stiffness decreases quickly to zeroat the end of unloading. However, the development of thestiffness from the beginning of reloading to the yield point(O2 − F2) corresponds to the elastic deformation and it canbe described with the model in Eq. (60) using the aboveobtained parameters: α = 2.0, spl,1 = 22µm. Due to hard-ening of the granules the slope of the curve at the secondcycle beyond the yield point (F2 − U2) is higher than at thefirst cycle.

The energy absorption during the static cyclic loading canbe quantitatively expressed by an equivalent restitution coef-ficient. By analogy with impact stressing, it is the squareroot of the ratio of elastic strain energy Wel released duringthe unloading to the total deformation energy WL , i.e. thework done by contact force during the compression of thegranule:

eeq = √Wel/WL (61)

Figure 28 shows the equivalent restitution coefficient of thegranules as a function of the number of cycles obtainedfrom the force-displacement curves of the above described

Fig. 27 Development of contact stiffness during the loading (◦),unloading (x) and reloading (�) of a zeolite 13X granule according tothe experiment in Fig. 21 (d1 = 1.68 mm, Fcyc = 7N, vB = 0.02mm/s)

Fig. 28 Effect of repeated one-way cyclic loading on the equivalentrestitution coefficient of the granules

cyclic tests. Due to the decrease of the deformation energyand energy absorption with increasing number of cycles, theequivalent restitution coefficient increases until a saturationstate in the plastic deformation. It is evident that the increaseof the restitution coefficient can also be shown during therepeated impact stressing of these granules on the same con-tact point. An example of this behaviour for brass sphereswas presented by Weir and Tallon [108]. However, the resti-tution coefficient of some materials decreases with increas-ing impact number due to the softening, microcracking andbreakage (see [92]).

123


Fig. 29 Theforce-displacement curvesduring the cyclic loading of azeolite 13X granule (d1 =1.62 mm, vB = 0.02 mm/s). Ateach maximum compressiveforce FU five cycles wereperformed. For clarity, only thefirst cycle of each series isshown

4.4 Dissipative behaviour during cyclic loadingwith increasing stressing intensity

In the previous section we discussed the behaviour of gran-ules at cyclic loading up to a maximum force, which was keptconstant in the test. In practice, the magnitude of the contactforce acting on a granule in a bed, e.g. during fluidisation,pneumatic transportation, mixing, shows a distribution thatdepends on the process parameters. An estimation of the evo-lution of the average interparticle force during bed settlingunder gravity after fluidisation was performed using DEM inthe publication of Moreno-Atanasio et al. [66].

In this section we describe how the behaviour of the gran-ules changes with increasing the maximum cyclic force. Agranule of zeolite 13X was first five times repeatedly com-pressed by a constant load amplitude of 2.2 N. The next fivecycles were performed at higher load amplitude of 5.2 N. Inthe same way, the next cycles were carried out at a force of8.2 N and subsequently at a force of 11.2 N.

Figure 29 shows the force-displacement curves of the firstcycles at the applied load amplitudes. With increasing num-ber of cycles the slope of the loading curve increases (pointU1 → U ′

1 in Fig. 29) and the amount of plastic deformationdecreases.

In the fifth cycle still a small plastic deformation occurs,since the zeolite 13X granules approach the saturation stateafter ∼15 cycles. When we increased the force above 2.2 Nin the sixth cycle the plastic deformation starts again whichwas detected after unloading by a significant increase inthe irreversible deformation in comparison to the previouscycle. The inflection point at the maximum force of previ-

Fig. 30 Effect of number of cycles on the equivalent restitution coef-ficient of zeolite 13X granules by increasing load amplitude

ous cycle (U ′1 in Fig. 29) indicates clearly the transition to

the plastic flow. During next cycles the slope of the loadingcurve changes only above this point.

It is noted here, that the loading curves of correspondingcycles at different load amplitudes are approximately paral-lel to each other in the plastic range, although four cyclesbetween these loadings were performed. Thus the contactstiffness of the plastic deformation is independent of the max-imum compressive force.

Furthermore, the curves showed that the previous cyclicloading at the lower force has no influence on the plasticdeformation and saturation state in following cycles at ahigher force. This assertion is confirmed by the equivalentcoefficient of restitution in Fig. 30(A, B) obtained from force-displacement curves (also see Table 6 for the first cycles).

123


Table 6 Energy balance of firstloading-unloading cycle atdifferent maximum forces

FU in N WLin µJ WL/m1 in J/kg Wel in µJ eeq

2.2 14.7 5.7 4.8 0.57

5.2 49.5 19.3 17.5 0.60

8.2 80.7 31.5 29.0 0.60

11.2 123.7 48.3 46.2 0.61

4.5 Free-fall tests

4.5.1 The coefficient of normal restitution

The coefficients of normal restitution of the granules wereobtained from the free-fall tests at different impact veloc-ities using above described device (Fig. 13). All granulesshow the elastic-plastic impact behaviour in the examinedvelocity range, Fig. 31. Furthermore, the following sequencefrom elastic-plastic to dominantly plastic behaviour (basedon the average coefficient of restitution) can be observed:γ-Al2O3 (en = 0.735), zeolite 4A (en = 0.653), zeolite13X (en = 0.644) and sodium benzoate (en = 0.532). Thesame sequence of behaviour from elastic to plastic range wasalso shown by these granules during compression tests.

The standard deviations of the restitution coefficients mea-sured by 50 repeated tests are relatively large. That wasexpected because of inhomogeneity of granules. For a givenparticle size, the contact stiffness and the position of theyield point are not constant, since the mechanical charac-teristics of the primary particles and the bonding agents arerandomly distributed within a fraction of granules. The indi-vidual granules differ in surface roughness and distribution ofsizes of pores. The standard deviation of the restitution coef-ficient corresponds to the standard deviation of contact stiff-ness obtained by compression tests of corresponding gran-ules (Table 3).

For all four examined granules the increasing impactvelocity in the examined range does not change the meannormal coefficient of restitution. In other words, the gran-ules exhibit elastic-plastic behaviour without a viscous effectduring the impact in this velocity range. As revealed previ-ously by loading-unloading tests of these granules the mag-nitude of maximum load does not affect the ratio of energyabsorption to the elastic strain energy. Under the assumptionthat the force-displacement behaviour during slow loadingis approximately the same as during dynamic stressing, thiscan give a rational explanation for constant restitution coef-ficient.

The measured restitution coefficient was fitted with themodel of Walton and Braun in Eq. (46), shown in Fig. 31as horizontal lines. Using the measured values of contactstiffness during loading (kL = kn,el,g for γ −Al2O3, zeolitesand kL = kn,pl,g for bigger fraction of sodium benzoate in

Fig. 31 Normal coefficient of restitution of examined granules versusimpact velocity

Table 3), the stiffness during the unloading was determinedas: kU = 3,368 N/mm for γ −Al2O3, kU = 2,094 N/mmfor zeolite 4A, kU = 636N/mm for zeolite 13X and kU =1,736 N/mm for sodium benzoate.

4.5.2 Maximum impact force of the elastic, elastic-plasticand plastic granules

From the free-fall experiments, we should estimate the max-imum impact forces that act on the granule by impact andcompare those with the yield point. Based on this informationthe conditions of the validity of Hertz theory to the performedimpacts can be examined.

In the case of the impact of an elastic granule on a rigidwall the Eq. (20) can be simplified as:

tel,cri t = tel

tw= 1.43

[m2

1 E1/21

(1 − v2

1

)2

v1 R61ρ

5/21

]1/5

. (62)

The dependence of the critical impact time tel,cri t on theimpact velocity v1 (Fig. 32) was calculated for three gran-ules using the material properties from Table 3. As has been

123


Fig. 32 Effect of the impact velocity v1 of the granules on the criticalimpact time tel,cri t

Fig. 33 Maximum displacement smax, maximum force Fel,max andduration tel of the impact versus impact velocity v1 during the elas-tic impact of γ-Al2O3granule on the steel wall

mentioned previously the elastic stress waves can show avisible influence only at high impact velocities (limv1→∞ tel,

cri t → 1). However, heretofore plastic deformation (pointF in Fig. 32) and breakage of granules occur. The minimumimpact velocity (v1,F in Fig. 32), at which the maximumimpact force exceeds the yield point of the granule and theplastic deformation is initiated, can be determined accordingto Davies [17]:

v1,F =(πpF

E∗)2

√2pF

5ρ1. (63)

Equations (15–17) can be used to calculate the maximumforce, maximum deformation and duration of impact forvelocities lower than v1,F where the Hertz theory is valid.Figure 33 shows an example of the calculation for theγ Al2O3 granules.

In the case of elastic-perfectly plastic deformation behav-iour of spherical granules, the maximum impact force can becalculated according to Thornton and Ning [94] as:

Fel,pl,max = FF

√√√√6

5

v21

v21,F

− 1

5, v1 ≥ v1,F . (64)

In this model the behaviour of the particle is assumed to beelastic at the beginning of loading and perfectly plastic afterexceeding the yield point (index el,pl in Eq. 64), where theforce FF at yield point is given by Eq. (28).

In the case of elastic-plastic deformation behaviour,beyond the yield point the centre of the contact area deformsplastically. But a ring at the periphery exhibits further elasticdeformation. The maximum impact force can be calculatedfor an adhesive particle [100]:

Fel−pl,max = π R∗ pF(κA − κp

)

×⎡⎣√(

κpaF=0

2κA

)2

+ 8ρ1 R∗2

3pF

(v2

1−v21,R

)+s2

F−κpaF=0

2κA

⎤⎦

−FH0, (65)

where κA is the elastic-plastic contact area coefficient givenin Eq. (23). v1 and v1,R are the impact and rebound veloci-ties, respectively. Neglecting the adhesion acting within thecontact zone (κp = 0, FH0 = 0) and sF (s2

F → 0) themaximum force is given by:

Fel−pl,max ≈ π R1 pFκA

√8ρ1 R2

1

3pF(1 − e)v1

= π R1 pFκAsel−pl,max. (66)

Figure 34 shows the maximum force as a function ofthe impact velocity calculated for different deformationbehaviour using models (16), (64) and (66). For this calcu-lation the mean values of the material properties of γ-Al2O3

and zeolite 13X in Table 3 were used.The yield point can be seen in Fig. 34 at the intersection of

the curves for elastic and elastic-perfectly plastic behaviourat the impact velocity of v1,F = 1.7 m/s for zeolite 13X andv1,F = 3.6 m/s for γ-Al2O3. Comparing these values withthe impact velocity of granules during the performed free-falltests (v1 = 0.5−4.5 m/s) it is found that the zeolite granulesexhibited the plastic deformation at drop heights above 0.15m and the impact of γ-Al2O3 granules after the fall fromthe height below 1.0 m had to behave elastic. However, themeasured restitution coefficients indicate the impact of thesegranules as elastic-plastic over the complete range of impactvelocities, i.e. the energy absorption occurs also at low veloc-ities and the yield point by collisional stressing is lower thanby slow compression.

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Fig. 34 The maximum impact force versus impact velocity at different deformation behaviour of the granules

Table 7 Comparison of energy absorption by compression and impact

Granules Diameter d1,50 in mm Impact Compression

enWdissWkin,1

en,eqWdissWL

γ-Al2O3 1.8 ± 0.05 0.735 ± 0.018 0.46 0.57 ± 0.02 0.68

zeolite 4A 2.3 ± 0.12 0.653 ± 0.007 0.57 0.50± 0.03 0.77

zeolite 13X 1.4 ± 0.05 0.644 ± 0.010 0.59 0.52 ± 0.06 0.72

sodium benzoate 1.7 ± 0.11 0.532 ± 0.018 0.72 0.11 ± 0.03 0.99

4.5.3 Comparison of the energy absorption by slowcompression and rapid impact loading

In comparison to the slow loading the same granules showhigher restitution coefficients during impact (Table 7). Dur-ing the impact the granules loose 15–27% of energy less thanduring compression. It is independent of the impact velocity(in the range of 0.5–4.5 m/s) or on the maximum compressionforce.

4.5.4 The tangential restitution coefficient

In addition to the normal restitution coefficient measurement,the velocity of granules before and after the oblique impacthas been measured with a high-speed camera to determinethe tangential restitution coefficient (Eq. 45). To evaluate theangular velocity ωR of granules at the rebound the granuleswere marked before testing, Fig. 35. The coefficient of tan-gential restitution was measured for γ-Al2O3, zeolite 4A andsodium benzoate granules. The zeolite 13X granules were notexamined as they have almost the same restitution coefficientas the zeolite 4A granules.

The coefficient of restitution in the normal direction doesnot depend on the impact angle in the range from 0◦ to 80◦in Fig. 36a. However, the coefficient of tangential restitutiondecreases with target inclination and reaches a minimum at an

Fig. 35 Images from the high-speed video recording of rebound of azeolite 4A granule (d = 2.3 mm) after the impact on steel wall

impact angle close to 30◦ (for γ-Al2O3 and zeolite 4A) and to20◦ (for sodium benzoate) Fig. 36b. The presence of minimain the function of tangential restitution coefficient is similar

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Fig. 36 Restitution coefficientin a normal and b tangentialdirection versus the impact anglefor three examined granules atimpact velocity of about 2.3 m/s

Fig. 37 Rebound angle (a) andangular velocity after impact (b)versus the impact angle for threeexamined granules at the impactvelocity of about 2.3 m/s

to that observed by Dong and Moys [18], Kharaz et al. [46]for different solid particles and it could be explained by tran-sition from rolling at small angles to sliding at large angles.

The increasing the inclination above the minimumincreases the coefficient of tangential restitution whichexceeds the normal coefficient of restitution and reaches avalue of 1 at angles close to 90◦ by glancing incidence.In the sliding regime (at impact angles higher 50◦) thetangential coefficient of restitution has the largest value forsodium benzoate and the least value for γ-Al2O3 as com-pared to the examined granules. Hence, this sequence isinversely to the coefficient of normal restitution of thesegranules.

Figure 37 summarizes the measurements of reboundangles and angular velocities of granules examined at impactangles from 0◦ to 80◦. When the impact takes place at anangle between 10◦ and 55◦, the rebound angle of γ-Al2O3

granules is slightly lower than the impact angle. The dot-ted line shows an ideal case of rotation at � = �R and thecoefficient of friction: µ →0. The shape of particles showedan important influence on the rebound behaviour at obliqueimpact. The small deviation from spherical shape for zeo-lite and great for sodium benzoate particles increases their

rebound angle, which is always higher than the impact angle,Fig. 37a. Moreover, the particle roughness influences the col-lision behaviour and therefore the rebound angle.

The angular velocity after rebound increases with increas-ing the impact angle, reaches a maximum at an angle ofabout 50◦ that shows the transition from rolling to sliding,Fig. 37b.

5 Conclusions

To simulate the rapid and collisional flow of particle bedsby DEM, the normal and tangential restitution coefficientsare usually required. In this work the mechanical behaviourand the associated restitution coefficients of various spheri-cal granules were determined through slow compression andrapid impact. In the case of compression, γ-Al2O3, zeolites13X and 4A granules show both elastic and elastic-plasticbehaviour. The Hertz theory predicts satisfactorily the elas-tic force-displacement behaviour of these granules. There-fore, Eq. (6) may be used in DEM with obtained modulusof elasticity and elastic contact stiffness. To describe theelastic-plastic non-adhesive deformation beyond the yieldpoint a force-displacement model (Eq. (25)) was developed.

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Characteristic parameters are the contact pressure and thedeformation at the yield point. These were determined usingmeasured force-displacement curves.

The contact stiffness of granules increases with increas-ing displacement in both the elastic and the elastic-plasticranges. Due to plastic deformation beyond yield point, theincrease of elastic-plastic stiffness with deformation is lessthan by elastic stiffness. Moreover, the increase in granulediameter leads to increase in stiffness.

The force-displacement curves of sodium benzoate gran-ules are linear, i.e. the stiffness is constant during plasticdeformation, which confirms the perfectly plastic model,Eq. (35). This model can be recommended for DEM as asimple force-displacement relationship for loading of ideallyplastic spherical granules. A good agreement between calcu-lated and measured radii of plastic deformed contact areawas obtained. The measured compressive strength and yieldpressure of plastic granules are independent of the granulesize.

Important effects of repeated loading on the deformationbehaviour of granules were described. The granules showhardening during cyclic stressing at constant load amplitude.However, the modulus of elasticity and the yield point of thegranule remain constant in each cycle but the elastic stiffnessincreases due to increase of the initial contact area. With acomfortable model (Eq. (59)) based on plastic deformationin the first cycle and a saturation parameter, the decrease andaccumulation of plastic deformation with increasing numberof cycles could be predicted. Moreover, based on the Hertzlaw an expression to calculate the elastic stiffness duringcycling loading was derived, Eq. (60). A good agreementbetween this model and the stiffness evolution in cyclic testsof zeolite 13X granules was obtained.

Moreover, the cyclic tests showed that hardening of gran-ules only takes place locally in the contact range wherethe maximum pressure during the compression occurs. Thelargest compaction of the microstructure occurs in the firstcycle due to plastic deformation.

After reaching certain saturation in the plastic deforma-tion, the granule behaves elastic-plastically and the restitu-tion coefficient remains constant. Therefore, in a fluidizedbed the effect of hardening due to repeated impacts of gran-ules on the magnitude of restitution coefficient will only berelevant during the first impacts of granules. As a first approx-imation of contact behaviour in DEM models, a constantvalue of the restitution coefficient can be assumed.

An additional effect on the cyclic hardening is the mag-nitude of maximum load. If this increase is before saturationthan the plastic deformation starts again, this significantlyreduces both the restitution coefficient and the contact stiff-ness.

The energy absorption during impact was found to belower than during compression, although the sequence of

granules from elastic-plastic to dominantly plastic behav-iour is the same in both stressing modes. Therefore, to pre-dict the energy absorption of granules during impact, thevalues of restitution coefficient obtained in free-fall exper-iments should be used in DEM. Moreover, the restitutioncoefficient of granules is independent of the impact velocityin the examined range of 0.5–4.5 m/s and of the maximumcompressive load. For DEM simulations of a fluidized bedwith dry granules, where the particle velocity is generally nothigher then 4.5 m/s, a constant coefficient of restitution cansafely be assumed.

From the study of oblique impact is found that the coeffi-cient of normal restitution is independent of the impact angle.Therefore, the tangential restitution coefficient shows mini-mum value at angles in the range of 20◦–30◦. The reboundangle of γ-Al2O3 granules is slightly lower (in the rangeof impact angles of 10◦–55◦) or equal to the impact angle.Because of the deviation from sphericity and high rough-ness the rebound angle of zeolite 4A and sodium benzo-ate granules is higher than the impact angle. The angularvelocity after rebound shows a maximum at an angle ofabout 50◦.

In future work, the obtained mechanical behaviour ofgranules will be used to select the appropriate contact modelfor their implementation in DEM and to analyze the dynam-ics of collisional particle flow in fluidized beds.

Acknowledgments The authors are grateful to the Institute of Exper-imental Physics, Otto-von-Guericke-University of Magdeburg for therent of the high-speed camera used in this study. We also thank M.Sc.L. Burai and M.Sc. S. Horbach for their help during performing of theexperiments.

Open Access This article is distributed under the terms of the CreativeCommons Attribution Noncommercial License which permits anynoncommercial use, distribution, and reproduction in any medium,provided the original author(s) and source are credited.

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